A finite element method for the solution of the time-dependent Maxwell equations in mixed form is presented. The method allows for local hp-refinement in space and in time. To this end, a space–time Galerkin approach is employed. In contrast to the space–time DG method introduced in [Van der Vegt, JCP(182) 2002] test and trial spaces do not coincide. This allows for obtaining a non-dissipative method. To obtain an efficient implementation, a hierarchical tensor product basis in space and time is proposed. This allows to evaluate the local residual with a complexity of \(\mathcal{O}(p^4)\) and \(\mathcal{O}(p^5)\) for affine and non-affine elements, respectively.

An hp-adaptive Discontinuous Galerkin Method for electromagnetic wave propagation phenomena in the time-domain is proposed. The method is highly efficient and allows for the first time the adaptive full-wave simulation of large, time-dependent problems in three-dimensional space. Refinement is performed anisotropically in the approximation order p and the mesh step size h regardless of the resulting level of hanging nodes. For guiding the adaptation process a variant of the concept of reference solutions with largely reduced computational costs is proposed. The computational mesh is adapted such that a given error tolerance is respected throughout the entire time-domain simulation.

Martin Lilienthal developed a space-time discontinuous Galerkin method. Unlike other works, this formulation is non-dissipative, which is achieved through non-coinciding choices of the trial and test spaces. Please find the preprint on arXiv.

This is another example of a dynamically adapted hp-mesh using SMOVE. The example is part of a paper I will submit soon (preprint link). It shows a triple slot patch antenna. The structure is included in the examples of CST MICROWAVE STUDIO.

The absolute value of the electric field is on the left and the hp-mesh using embedded triangles for visualizing the tensor product orders (cf. Demkowicz, Solin) is on the right. The adaptation strategy is HP_ANISO, i.e., allowing for anisotropic refinement in the mesh step size and the polynomial order. The computation and adaptation is done in a three-dimensional domain. The viewplane is located at the bottom of the antenna substrate.